Template:Bounds on time given instantaneous mtbf camsaa-gd: Difference between revisions

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(Created page with '===Bounds on Time Given Instantaneous MTBF=== ====Fisher Matrix Bounds==== The time, <math>T</math> , must be positive, thus <math>\ln T</math> is treated as being normally di…')
 
 
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===Bounds on Time Given Instantaneous MTBF===
#REDIRECT [[Crow-AMSAA_-_NHPP#Bounds_on_Time_Given_Instantaneous_MTBF_2]]
====Fisher Matrix Bounds====
The time,  <math>T</math> , must be positive, thus  <math>\ln T</math>  is treated as being normally distributed.
 
::<math>\frac{\ln \hat{T}-\ln T}{\sqrt{Var(\ln \hat{T}})}\ \tilde{\ }\ N(0,1)</math>
 
Confidence bounds on the time are given by:
 
::<math>CB=\hat{T}{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{T})}/\hat{T}}}</math>
 
:where:
 
::<math>\begin{align}
  & Var(\hat{T})= & {{\left( \frac{\partial T}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial T}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\
&  & +2\left( \frac{\partial T}{\partial \beta } \right)\left( \frac{\partial T}{\partial \lambda } \right)cov(\hat{\beta },\,\,\,\hat{\lambda }) 
\end{align}</math>
 
The variance calculation is the same as Eqn. (variances) and:
 
::<math>\hat{T}={{(\lambda \beta \cdot {{m}_{i}}(T))}^{1/(1-\beta )}}</math>
 
::<math>\begin{align}
  & \frac{\partial T}{\partial \beta }= & {{\left( \lambda \beta \cdot \text{ }{{m}_{i}}(T) \right)}^{1/(1-\beta )}}\left[ \frac{1}{{{(1-\beta )}^{2}}}\ln (\lambda \beta \cdot {{m}_{i}}(T))+\frac{1}{\beta (1-\beta )} \right] \\
& \frac{\partial T}{\partial \lambda }= & \frac{{{(\lambda \beta \cdot \text{ }{{m}_{i}}(T))}^{1/(1-\beta )}}}{\lambda (1-\beta )} 
\end{align}</math>
 
====Crow Bounds====
:Step 1: Calculate the confidence bounds on the instantaneous MTBF:
 
::<math>MTB{{F}_{i}}={{\widehat{m}}_{i}}(1\pm W)</math>
 
:Step 2: Use equations in 5.4.5.2 to calculate the time given the instantaneous MTBF.

Latest revision as of 03:51, 24 August 2012

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